Question: Let $F_n$ be the Fibonacci sequence, that is, $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$.  Compute
\[\sum_{n=0}^{\infty} \frac{F_n}{10^n}.\]
Let $S = \sum_{n = 0}^\infty \frac{F_n}{10^n}.$  Then
\begin{align*}
S &= F_0 + \frac{F_1}{10} + \frac{F_2}{10^2} + \frac{F_3}{10^3} + \dotsb \\
&= \frac{F_0 + 1}{10} + \frac{F_1 + F_0}{10^2} + \frac{F_2 + F_1}{10^3} + \dotsb \\
&= \frac{1}{10} + \frac{F_0}{10} + \frac{F_1}{10^2} + \frac{F_2}{10^3} + \dotsb + \frac{F_0}{10^2} + \frac{F_1}{10^3} + \dotsb \\
&= \frac{1}{10} + \frac{1}{10} S + \frac{1}{10^2} S.
\end{align*}Solving, we find $S = \boxed{\frac{10}{89}}.$